### About Negative Exponents & the Quotient Rule:

When we simplify expressions that contain exponents, we often have to utilize the quotient rule for exponents. Additionally, we will often come across many scenarios where we need to simplify an expression with negative exponents or an exponent of zero.

Test Objectives
• Demonstrate a general understanding of the rules of exponents
• Demonstrate the ability to simplify an expression using the quotient rule for exponents
• Demonstrate the ability to simplify an expression with negative exponents
• Demonstrate the ability to simplify an expression with an exponent of zero
Negative Exponents & the Quotient Rule Practice Test:

#1:

Instructions: Simplify each.

$$a)\hspace{.2em}\frac{2x^3y^2}{2^{-1}x^2y^5}$$

$$b)\hspace{.2em}\frac{(2^{-2}x^4y^{-2})^{-1}}{2x^{-3}y^7}$$

#2:

Instructions: Simplify each.

$$a)\hspace{.2em}\frac{2^3x^5y^{-3}}{(2^{-3}x^5y^{-6})^{-2}}$$

$$b)\hspace{.2em}\frac{x^5y^6z^4}{x^{-5}y^{-6}z^{-4}}$$

#3:

Instructions: Simplify each.

$$a)\hspace{.2em}\frac{x^9y^7z^{-10}}{xy^{-5}z^{11}}$$

$$b)\hspace{.2em}\frac{(x^5y^2)^{-3}}{x^4y^4z^{-2}}$$

#4:

Instructions: Simplify each.

$$a)\hspace{.2em}\frac{(x^2y^{-4}z)^{-5}}{x^4y^9z^{11}}\cdot \frac{1}{(x^4y^9z^{11})^{-1}}$$

$$b)\hspace{.2em}\frac{(xyz)^{-4}}{(x^5y^3z)^{-3}}\cdot \frac{(x^7y^4z^2)^{-2}}{(x^4y^4z^4)^{-1}}$$

#5:

Instructions: Simplify each.

$$a)\hspace{.2em}\frac{(x^9y^{12}z^2)^0}{x^{-9}y^{-12}z^{-2}}\cdot \frac{(x^{-5}z^2z^{15})^{-3}}{xyz^5}$$

$$b)\hspace{.2em}\frac{(2y^3z^4)^{-2}\cdot (2yz^5)^2}{3^2(y^4z)^3 \cdot (3y^3z^2x)^{-4}}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}\frac{2^2x}{y^3}$$

$$b)\hspace{.2em}\frac{2}{xy^5}$$

#2:

Solutions:

$$a)\hspace{.2em}\frac{x^{15}}{2^3y^{15}}$$

$$b)\hspace{.2em}x^{10}y^{12}z^8$$

#3:

Solutions:

$$a)\hspace{.2em}\frac{x^8y^{12}}{z^{21}}$$

$$b)\hspace{.2em}\frac{z^2}{x^{19}y^{10}}$$

#4:

Solutions:

$$a)\hspace{.2em}\frac{y^{20}}{x^{10}z^5}$$

$$b)\hspace{.2em}\frac{xy}{z}$$

#5:

Solutions:

$$a)\hspace{.2em}\frac{x^{23}y^{11}}{z^{54}}$$

$$b)\hspace{.2em}\frac{9x^4z^7}{y^4}$$